We extend some classical statistical inference to the case of a random number of observations with a stabilized distribution: namely, in the normal model, inference for the mean with known and unknown variance and inference for the variance. We describe some useful models for the number of observations obtained by truncation or translation of usual models given by integer-valued random variables: Poisson, binomial, geometric, and negative binomial. We present an efficient random search algorithm for the computation of the quantiles of the relevant statistics, we describe an interval estimation procedure for the extended model, and we propose a parametric bootstrap simulation study to validate the proposed procedure.

The geometric Brownian motion (GBM) is very popular in modeling the dynamics of stock prices. However, the constant volatility assumption is questionable and many models with nonconstant volatility have been developed. In the papers [ 7 M.L. Esquível and P.P. Mota, On some auto-induced regime switching double-threshold glued diffusions, J. Stat. Theory Pract. 8 (2014), pp. 760–771. doi: 10.1080/15598608.2013.854184. [Taylor & Francis Online] [Google Scholar] , 12 P. P. Mota and M.L. Esquível, On a continuous time stock price model with regime switching, delay, and threshold, Quant. Financ. 14 (2014), pp. 1479–1488. doi: 10.1080/14697688.2013.879990. [Taylor & Francis Online], [Web of Science ®] [Google Scholar] ] the authors introduce a regime switching process where in each regime the process is driven by GBM and the change in regime is defined by the crossing of a threshold. In this paper we used Akaike's and Bayesian information criteria to show that the GBM with regimes provides a better fit than the GBM. We also perform a forecasting comparison of the models for two selected companies.

Discussiones Mathematicae Probability and Statistics